| L(s) = 1 | − 3-s − 4·5-s + 2·7-s + 9-s + 2·11-s + 4·15-s + 6·17-s − 2·19-s − 2·21-s − 8·23-s + 11·25-s − 27-s − 6·29-s − 10·31-s − 2·33-s − 8·35-s − 4·37-s − 4·43-s − 4·45-s + 2·47-s − 3·49-s − 6·51-s + 10·53-s − 8·55-s + 2·57-s − 14·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 1.03·15-s + 1.45·17-s − 0.458·19-s − 0.436·21-s − 1.66·23-s + 11/5·25-s − 0.192·27-s − 1.11·29-s − 1.79·31-s − 0.348·33-s − 1.35·35-s − 0.657·37-s − 0.609·43-s − 0.596·45-s + 0.291·47-s − 3/7·49-s − 0.840·51-s + 1.37·53-s − 1.07·55-s + 0.264·57-s − 1.82·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6843418918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6843418918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.09833083932010, −14.56462508257559, −14.24493674918238, −13.44500771938806, −12.54496563755953, −12.30248740940424, −11.87298816035802, −11.39033657764180, −10.94292177172952, −10.45105455038459, −9.712410401388508, −9.022077548307176, −8.375343154078622, −7.821836673483235, −7.546447121954032, −6.954383378142496, −6.175682399618365, −5.476597258004094, −4.970156101558116, −4.156224165697861, −3.786321446567882, −3.337280317745270, −2.038288367348968, −1.348400028367882, −0.3440503664316052,
0.3440503664316052, 1.348400028367882, 2.038288367348968, 3.337280317745270, 3.786321446567882, 4.156224165697861, 4.970156101558116, 5.476597258004094, 6.175682399618365, 6.954383378142496, 7.546447121954032, 7.821836673483235, 8.375343154078622, 9.022077548307176, 9.712410401388508, 10.45105455038459, 10.94292177172952, 11.39033657764180, 11.87298816035802, 12.30248740940424, 12.54496563755953, 13.44500771938806, 14.24493674918238, 14.56462508257559, 15.09833083932010
